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We prove that, even using local information, the proposed framework asymptotically converges to a desired swarm distribution and it retains the advantages of existing closed-loop-type approaches. This paper explicitly presents the design requirements for a time-inhomogeneous Markov chain to achieve these desired featur... |
). The rest of this paper are organised as follows. Section II introduces the desired features of a swarm distribution guidance framework along with relevant definitions and notations. Section III proposes our framework with its design requirements, the biological inspiration, and an analysis regarding whether the desi... |
Notations [MATH] [MATH] and denote the empty set, the zero matrix of appropriate sizes, the identity matrix of appropriate sizes, and a row vector with all elements are equal to one, respectively. |
[MATH] is a stochastic (row) vector such that [MATH] and [MATH] [MATH] indicates the [MATH] -th element of vector [MATH] [MATH] denotes the probability that event [MATH] will happen. |
II Preliminaries II-A Definitions This section presents necessary definitions and assumptions for our proposed framework, which will be shown in Section III Since most of them are embraced from the recent existing literature |
, we here briefly provide their essential meanings. Definition 1 Agents and Bins A set of agents [MATH] are supposed to be distributed over a prescribed region in a state space [MATH] . The entire space is partitioned into [MATH] disjoint bins (subspaces) such that |
[MATH] and [MATH] [MATH] We also regard [MATH] as the set of all the bins. Each bin [MATH] represents a predefined range of an agent’s state, e.g., position. The number of the entire agents is time-varying, and its value at time instant [MATH] is denoted by [MATH] Note that we do not assume that the agents keep track o... |
Definition 2 Agent’s state Let [MATH] be the state indicator vector of agent [MATH] at time instant [MATH] If the agent’s state belongs to bin [MATH] , then [MATH] , otherwise [MATH] |
Definition 3 Current (global) swarm distribution The current (global) swarm distribution [MATH] is a row-stochastic vector such that each element [MATH] is the population fraction (swarm density) of [MATH] in bin [MATH] at time instant [MATH] |
[EQUATION] Definition 4 Agent’s stochastic state and decision policy Agent [MATH] ’s stochastic state is a row-stochastic vector [MATH] |
in which each element [MATH] gives the probability that the agent’s state belongs to bin [MATH] at time instant [MATH] [EQUATION] |
The probability that agent [MATH] in bin [MATH] at time instant [MATH] will transition to bin [MATH] before the next time instant is called its stochastic decision policy , denoted as: |
[EQUATION] Note that [MATH] is a row-stochastic matrix such that [MATH] and [MATH] , and will be referred as Markov matrix Definition 5 |
Desired swarm distribution The desired swarm distribution [MATH] is a row-stochastic vector such that each element [MATH] indicates the desired swarm density for bin [MATH] |
Assumption 1 For ease of description for this paper, we assume that [MATH] [MATH] Obviously, in practice, there may exist some bins whose desired swarm densities are zero. These bins can be accommodated by adopting any subroutines ensuring that all agents eventually move to and remain in any of the positive-desired-den... |
Assumption 2 The number of agents It is assumed that [MATH] so that the time evolution of the swarm distribution is governed by the stochastic decision policy in Equation ( ). Although the finite cardinality of the agents normally cause a residual convergence error, a lower bound on [MATH] that probabilistically guaran... |
Definition 6 Physical motion constraint Motion constraints of agents are denoted by the matrix [MATH] where [MATH] if agents in bin [MATH] at time instant [MATH] are allowed to transition to bin [MATH] by the next time instant; |
[MATH] , otherwise. It is assumed that [MATH] is symmetric and irreducible (i.e., strongly-connected); and [MATH] for all agents, bins, and time instants. |
Definition 7 Communicationally-connected Bins [MATH] and [MATH] are said to be communicationally-connected if there exists at least one agent in bin [MATH] who can directly communicate with some agents in bin [MATH] , and vice versa. This communicational connectivity over all the bins at time instant [MATH] is defined ... |
Assumption 3 Communicational connectivity over bins The physical motion constraint of a robot is, in general, more stringent than its communicational constraint. From this, it can be assumed that if the transition of agents between bin [MATH] and [MATH] is allowed within a unit time instant, then the both bins are comm... |
Using distributed consensus algorithms each agent can access necessary local information in its neighbour bins. Assumption 4 Pre-known Information |
The desired swarm distribution [MATH] , the motion constraint matrix [MATH] (also [MATH] ), and other pre-determined values such as variables regarding objective functions and user-design parameters (which will be introduced later) are known by all the agents before they begin a mission. |
Assumption 5 Agent’s capability Each agent can determine the bin to which it belongs, and know the locations of neighbour bins so that it can navigate toward any of these bins. The agent is capable of collision avoidance behaviours against other agents or obstacles. |
II-B Problem Statement The objective of the swarm distribution guidance problem considered in this paper is to distribute a set of agents [MATH] over a set of bins [MATH] by the Markov matrix [MATH] in a manner that holds the following desired features: |
Desired Feature 1 The swarm distribution [MATH] asymptotically converges to the desired swarm distribution [MATH] as time instant [MATH] goes to infinity. |
Desired Feature 2 Transitions of the agents between the bins are controlled in a way that [MATH] becomes close to [MATH] as [MATH] converges to [MATH] . This implies that the agents are settled down after [MATH] is achieved, and thus unnecessary transitions can be reduced. Moreover, the agents identify and compensate a... |
Desired Feature 3 For each agent in bin [MATH] , the information required for generating time-varying stochastic decision policies is not global information (e.g., [MATH] ) but locally available information within [MATH] Thereby, the resultant time-inhomogeneous Markov process is based on LICA, and has benefits such as... |
Remark 1 One of our main contributions is to provide Desired Feature as well as to retain Desired Features and by additionally adopting Assumption , which can be elicited from other assumptions in the existing literature. |
III A Closed-loop-type Framework using Local Information This section proposes a LICA-based framework for the swarm distribution guidance problem. The framework is different from the recent closed-loop-type algorithms in |
in the sense that they utilise the global information (e.g., the current swarm distribution in Equation ( )) for constructing a time-inhomogeneous Markov matrix, whereas ours uses the local information in Equation ( ). We present, in spite of using such relatively insufficient information, how the desired features desc... |
III-A The Biological Inspiration For a swarm of fishes, it has commonly been assumed that their crowdedness limits their perception ranges over other members, and their cardinality restricts the capacity for individual recognition |
How fishes end up with collective behaviours is different from the ways of other social species such as bees and ants, which are known to use recruitment signals for the guidance of the entire swarm |
Thus, in biology domain, a question naturally has arisen about the mechanism of fishes’ decision-making in an environment where local information is only available and information transfer between members does not explicitly happen |
It has been experimentally shown that fishes’ swimming activities vary depending on their perceivable neighbours. According to , fishes have the tendency to maintain their statuses (e.g., position, speed, and heading angle) relative to those of other nearby fishes, which results in their organised formation structures.... |
that spatial density of fishes has influences on both the minimum distances between them and the primary orientation of the fish school. |
Based on this knowledge, the works in suggest individual-based models to further understand the collective behavioural mechanisms of fishes: for example, their repelling, attracting, and orientating behaviours |
how the density of informed fishes affects the elongation of the formation structure and group-size choices The common and fundamental characteristic of these models is that every agent maintains or adjusts its personal status with consideration of those of other individuals within its limited perception range. |
As inspired by the understanding of fishes, we believe that there must be an enhanced swarm distribution guidance approach in which each agent only needs to keep its relative status by using local information available from its nearby neighbours. In this approach, a global information is not necessary to be known by ag... |
III-B Fundamental Idea of the Proposed Approach Suppose that each agent in bin [MATH] is required to keep its local status [MATH] , which we referred to as the current local swarm density at bin [MATH] , at the value of the corresponding locally-desired swarm density |
[MATH] . They are respectively defined as follows: [EQUATION] where [MATH] is the number of agents such that [MATH] ; and [EQUATION] |
We use the term [MATH] as an estimate of [MATH] by agent [MATH] , which can be obtained through a distributed information consensus algorithm |
The fundamental idea of the proposed approach is to make each agent [MATH] in bin [MATH] (i) only need to estimate the difference of [MATH] and [MATH] , which are both locally-available information within [MATH] ; and |
(ii) more reluctant to deviate from the current bin as the difference becomes smaller (i.e., [MATH] and [MATH] as [MATH] ). Our proposed framework utilises the difference between [MATH] and [MATH] as a local-information-based feedback gain, denoted by [MATH] , which is a scalar in [MATH] that monotonically decreases as... |
[EQUATION] where [MATH] and [MATH] are design parameters. We call this gain primary local-feedback gain because it is utilised to control the primary guidance matrix [MATH] (shown in the next subsection). |
Remark 2 Equation ( ) is equivalent to the [MATH] -th element of the following vector: [EQUATION] Namely, [MATH] if [MATH] Here, we intentionally introduce Equation ( ) for ease of comparison with the information required for feedback gains in the existing literature (e.g., Equation ( )). From this, it is implied that,... |
III-C A LICA-based Closed-loop-type Framework This subsection presents our closed-loop-type framework based on locally-available information feedbacks. The basic form of the stochastic decision policy for agent [MATH] in bin [MATH] is such that |
[EQUATION] Here, [MATH] is the weighting factor to have different weights on the agent’s primary decision policy [MATH] and secondary decision policy [MATH] . It is defined as |
[EQUATION] where [MATH] is a design parameter; and [MATH] is secondary local-feedback gain , which is based on the difference between [MATH] and [MATH] Note that [MATH] is mainly affected by [MATH] , while diminishing as time instant [MATH] goes to infinity. |
Equation ( ) can be represented in matrix form as [EQUATION] where [MATH] and [MATH] are row-stochastic matrices, called primary guidance matrix and secondary guidance matrix , respectively. |
[MATH] is a diagonal matrix such that [MATH] The stochastic state vector of agent [MATH] is governed by the Markov process: [EQUATION] |
For now, we claim that, in order for this Markov system to achieve Desired Features [MATH] must satisfy the following requirements. |
Requirement 1 [MATH] is a matrix with row sums equal to one, i.e., [EQUATION] In fact, [MATH] needs to be row-stochastic, for which it should further hold that [MATH] [MATH] . Note that this constraint is implied by ( R4 ), which will be introduced later. |
Requirement 2 All diagonal elements are positive, i.e., [EQUATION] Requirement 3 The stationary distribution of [MATH] is the desired swarm distribution [MATH] , i.e., |
[EQUATION] With consideration of ( R1 ), this requirement can be fulfilled by [MATH] [MATH] A Markov process satisfying this property is said to be reversible |
Requirement 4 [MATH] is irreducible such that [EQUATION] Note that [MATH] is already assumed to be irreducible in Assumption Requirement 5 |
[MATH] becomes close to [MATH] as [MATH] converges to [MATH] , i.e., [EQUATION] Depending on the objectives of a user, [MATH] [MATH] [MATH] and [MATH] can be designed differently under given specific constraints. As long as [MATH] holds ( R1 )-( R5 ) for all time instant [MATH] and all agent [MATH] , the aforementioned... |
Every agent executes the following algorithm at every time instant. The detail regarding Line 10 will be presented in Section IV , which shows examples of how to implement this framework. |
Algorithm 1 Decision making of agent [MATH] at time instant [MATH] 1: // Obtain the local information 2: Identify the current bin [MATH] |
3: Identify neighbour bins [MATH] (and [MATH] ); 4: Compute [MATH] using ( ); 5: Obtain [MATH] 6: // Generate the stochastic decision policy |
7: Compute [MATH] (using ( )); 8: Compute [MATH] [MATH] 9: Compute [MATH] [MATH] 10: Compute [MATH] 11: Compute [MATH] using ( ); |
12: Compute [MATH] [MATH] using ( ); 13: // Individually behave based on the policy 14: Generate a random number [MATH] 15: Select bin [MATH] such that |
16: [MATH] 17: Move to the selected bin; III-D Analysis We first show that the Markov process in Equation ( 11 ) holds Desired Feature under the assumption that [MATH] satisfies the requirements ( R1 )-( R4 ) for each time instant. The stochastic state of agent [MATH] at time instant [MATH] governed by the Markov proce... |
[EQUATION] For ease of analysis, we assume that every agent [MATH] knows any necessary information correctly, i.e., [MATH] Theorem 1 |
Provided that the requirements ( R1 )-( R4 ) are satisfied for all time instants [MATH] it holds that [MATH] pointwise for all agents, irrespective of the initial condition. |
Proof. This claim can be proved by following similar steps in proving 13 , Theorem 4] The claim is true if [MATH] In order for that, the matrix product [MATH] should (i) be strongly ergodic and (ii) have [MATH] as its unique limit vector, i.e., [MATH] We will show that the two conditions are valid under the assumption ... |
Lemma in Appendix describes the characteristics of [MATH] and [MATH] , which will be used for the rest of this proof. From this lemma, (a) [MATH] is primitive (thus, regular); (b) there exists a positive lower bound [MATH] for [MATH] [MATH] ; and (c) [MATH] is asymptotically homogeneous. Then, from 33 , Theorem 4.15, p... |
Let [MATH] be the unique stationary distribution vector corresponding to [MATH] (i.e., [MATH] ). Due to the prior condition (b) and the fact that (d) [MATH] is irreducible for [MATH] it follows from 33 , Theorem 4.12, p.149] that the asymptotical homogeneity of [MATH] with respect to [MATH] (i.e., [MATH] ) is equivalen... |
[MATH] and [MATH] , where is a limit vector. According to 33 , Corollary, p.150] , under the prior conditions (b) and (d), if [MATH] is strongly ergodic with its unique limit vector , then [MATH] Hence, it turns out that the unique limit vector of [MATH] is [MATH] (i.e, [MATH] ). Thereby, the condition (ii) is also ful... |
Theorem implies that the stochastic state of any agent eventually converges to the desired swarm distribution, regardless of [MATH] [MATH] and ( R5 ). In other words, even if ( R5 ) is not satisfied, the Markov system can converge to [MATH] However, the system induces unnecessary transitions of agents even after being ... |
For now, we present that Desired Feature can be obtained by ( R5 ) and Theorem , which will be described later. Suppose that, for every bin [MATH] [MATH] converges to and eventually reaches [MATH] at some time instant [MATH] The following shows that at this moment it also holds that [MATH] reaches [MATH] . From Equatio... |
[MATH] , it follows that [MATH] [MATH] This can be rearranged as: [EQUATION] where [MATH] is a diagonal matrix such that [MATH] [MATH] is the communicational connectivity matrix (in Definition ); and [MATH] is a row vector such that the [MATH] -th element indicates [MATH] , i.e., the number of agents in bin [MATH] at t... |
Lemma 1 Given [MATH] bins communicationally-connected as a tree-type topology, the rank of its corresponding matrix [MATH] in Equation ( 13 ) is [MATH] |
Proof. The matrix [MATH] can be linearly decomposed into [MATH] of the same-sized matrices [MATH] , where [MATH] is the number of edges in the underlying graph of [MATH] Here, [MATH] is a matrix such that |
[MATH] and [MATH] [MATH] ; and all the other entries are zero. For example, consider that four bins are given and connected as shown in Figure (a). Clearly, [MATH] , where |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] It is trivial that the rank of every [MATH] is one, and the matrix has only one linearly independent column vector, denoted by [MATH] . Without loss of generality, we consider [MATH] as a column vector such that the [MATH] -th entry is [MATH] , the [MATH] -th entry is [MATH] ... |
It is obvious that [MATH] and [MATH] are linearly independent when the bin pairs [MATH] and [MATH] are different. This implies that the number of linearly independent column vectors of [MATH] is the same as that of edges in the topology. Hence, for a tree-type topology of [MATH] bins, since there exist [MATH] edges, th... |
Lemma 2 Given a strongly-connected topology of bins, the rank of its corresponding matrix [MATH] is not affected by adding a new edge that directly connects any two existing bins. |
Proof. We will show that this claim is valid even when a tree-type topology is given, as it is a sufficient condition for strong-connectivity. Given the tree-type topology in Figure (a), suppose that bin [MATH] and [MATH] are newly connected. Then, the new topology becomes as shown in Figure (b), and it has new corresp... |
Thanks to Lemma and we end up with the following corollary and theorem: Corollary 1 Given [MATH] bins that are communicationally strongly-connected, the rank of its corresponding [MATH] is [MATH] |
Theorem 2 Given [MATH] bins that are communicationally strongly-connected, convergence of [MATH] to [MATH] is equivalent to convergence of [MATH] to [MATH] |
Proof. From Equation ( ), it can be said that [MATH] When [MATH] is assumed to converge to [MATH] at some time instant [MATH] for every bin [MATH] Equation ( 13 ) is valid (i.e., [MATH] ). Since the nullity of [MATH] is one, due to Corollary , there is only one linearly-independent row-vector [MATH] such that [MATH] He... |
From this theorem and ( R5 ), Desired Feature finally holds. Corollary 2 If [MATH] satisfies ( R5 ), it can be said from Theorem that [MATH] becomes [MATH] as [MATH] converges to [MATH] And this is also the case for the Markov process [MATH] , which satisfies Desired Feature |
In order for each agent [MATH] in bin [MATH] to generate the time-varying stochastic decision policy [MATH] in Equation ( ), the agent only needs to obtain its local information within [MATH] . Therefore, Desired Feature is also achieved. |
Remark 3 Robustness against dynamic changes of agents and those of bins The proposed framework is robust with against dynamic changes in the number of agents and bins. Similarly to what is claimed in 13 , Remark 8] , as each agent behaves based on its current bin location and local information in a memoryless manner, D... |
IV Implementation Examples IV-A Example I: Minimising Travelling Expenses This section provides examples on implementations of the framework proposed. In particular, this subsection addresses a problem of minimising travelling expenses of agents during convergence to a desired swarm distribution. |
This problem can be defined as: given a cost matrix [MATH] in which each element [MATH] represents the travelling expense of an agent from bin [MATH] to [MATH] , find [MATH] such that |
[EQUATION] subject to ( R1 )-( R5 ) and [EQUATION] where [MATH] is a design parameter. [MATH] is set by [EQUATION] so that the value monotonically increases with regard to increase of either [MATH] or [MATH] and diminishes as [MATH] and [MATH] simultaneously reduces. This value controls the lower bound of [MATH] in Equ... |
[MATH] is a scalar that monotonically decreases as [MATH] increases (see Equation ( 29 ) for instances), encouraging agents in bin [MATH] to avoid spending higher transition expenses. Note that we assume that [MATH] is symmetric; [MATH] if [MATH] ; and its diagonal entries are zero. |
Corollary 3 The optimal matrix [MATH] of the problem ( P1 ) is given by: [MATH] and [MATH] [EQUATION] and [MATH] [EQUATION] Proof. |
We can prove this by following the proof of 13 , Corollary 1] Suppose that the problem is only subject to ( R4 ) and ( 14 ), without ( R1 )-( R3 ) and ( R5 ). Then, the off-diagonal elements of an optimal matrix should be their corresponding lower bounds in ( 14 ) if [MATH] The diagonal elements of the matrix do not af... |
Let us now consider ( R1 )-( R3 ) and ( R5 ). Since [MATH] [MATH] and [MATH] are upper-bounded by [MATH] and [MATH] [MATH] in ( 17 ) is always positive for all [MATH] , which fulfils ( R2 ). It is also obvious that ( R1 ) is satisfied by Equation ( 17 ). From Equation ( 16 ), it holds that [MATH] , complying with ( R3 ... |
For reducing unnecessary transitions of agents during this process, it is favourable that agents in bin [MATH] such that [MATH] (i.e., underpopulated) do not deviate. To this end, we set |
[MATH] and [MATH] as follows [EQUATION] The gain value is depicted in Figure (a) with regard to [MATH] Remark 4 Increase of Convergence Rate |
Due to the fact that [MATH] from Equation ( 16 ), the total outflux of agents from bin [MATH] becomes smaller as the bin has fewer connections with other bins. This eventually makes the convergence rate of the Markov process slower. |
Adding an additional variable into [MATH] in ( 16 ) does not affect the obtainment of Desired Features as long as [MATH] satisfies ( R1 )-( R5 ). Thus, in order to enhance the convergence rate under the requirements, one can add |
[EQUATION] into [MATH] , as follows: [EQUATION] which can be substituted for Equation ( 16 ). Algorithm 2 Subroutine of Algorithm (Line 10 ) for P1 |
1: Compute [MATH] [MATH] using ( 16 ) (or ( 20 )) and ( 17 ); 2: Set [MATH] and [MATH] 3: Compute [MATH] using ( 18 ); IV-B Example II: Maximising Convergence Rate within Upper Flux Bounds |
This subsection presents an example in which the specific objective is to maximise the convergence rate under upper bounds regarding transitions of agents between bins, denoted by upper flux bounds The bounds can be interpreted as safety constraints in terms of collision avoidance and congestion: higher congestions may... |
, where transitions of agents are limited only at a desired swarm distribution. This restriction is not for considering the aforementioned safety constraints, but rather for mitigating the trade-off between convergence rate and long-term system efficiency. |
For the sake of imposing upper flux bounds during the entire process, we consider the following one-way flux constraint: [EQUATION] |
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